Topological phases protected by reflection symmetry and cross-cap states

Twisting symmetries provides an efficient method to diagnose symmetry-protected topological (SPT) phases. In this paper, edge theories of (2+1)-dimensional topological phases protected by reflection as well as other symmetries are studied by twisting reflection symmetry, which effectively puts the edge theories on an unoriented space-time, such as the Klein bottle. A key technical step taken in this paper is the use of the so-called cross-cap states, which encode entirely the unoriented nature of space-time, and can be obtained by rearranging the space-time geometry and exchanging the role of space and time coordinates. When the system is in a nontrivial SPT phase, we find that the corresponding cross-cap state is noninvariant under the action of the symmetries of the SPT phase, but acquires an anomalous phase. This anomalous phase, with a proper definition of a reference state, on which symmetry acts trivially, reproduces the known classification of (2+1)-dimensional bosonic and fermionic SPT phases protected by reflection symmetry, including in particular the ${\mathbb{Z}}_8$ classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.

Figure 1

Rearrangement of space-time. (a) The original loop channel (Euclidean) space-time. Here the points $(A,B)$ and $(A^{\prime },B^{\prime })$ in the space-time are identified due to the boundary conditions Eq. (12). The line segments with the same symbols are also identified by the boundary conditions Eq. (12). (b) We shift the box of the space-time formed by $A^{\prime }{B}^{\prime }{D}^{\prime }{C}^{\prime }$ along $x_2$ by $\beta$. (c) After shifting the box, we flip the orientation of the box with respect to $x_1=3\ell /4$. (d) We slide the box along $x_1$ by $\ell /2$. Then we obtain the cross-cap geometry. Notice that any $x_2\in [0,\beta ]$ is identified with its antipodal point at the slice of “time” at $x_1=0$ and $x_1=\ell /2$. By renaming the variables $x_1\to {\sigma }_1$ and $x_2\to {\sigma }_2$, we obtain the space-time of the tree channel. Here the ${\sigma }_1$ direction is taken to be a direction of (fictitious) time evolution. At the boundary of space-time located at ${\sigma }_1=0$ and ${\sigma }_1=\ell /2$, there are cross-caps.